Accurate distributions of Mallows' Cp and its unbiased modifications with applications to shrinkage estimation

•The accurate distributions of Mallows' Cp and its modifications are given.•The shrinkage estimators corresponding to Cp and its modifications are proposed.•The shrinkage estimators show better behaviors than the current ones.

In multivariate multiple linear regression with normal errors, the statistics corresponding to Mallows' Cp and its unbiased estimators are dealt with. When a model is not underfitted, the expectations and variances of the statistics are given. In the special case of univariate multiple regression, when a model may or may not be underfitted, accurate distributions of the statistics are obtained as linear functions of a noncentral F distribution. Using these results, it is possible to obtain the multiplicative constants for unbiased estimators, while minimizing their mean square errors. Simulations for model selection show that shrinkage estimators that use the minimizing constants give a substantially higher proportion of selecting correct models than the estimators using current criteria under normality and non-normality.